$G_\delta $ and co-meager semifilters

William R. Brian¹ ; Jonathan L. Verner²

¹ Department of Mathematics Tulane University 6823 St. Charles Ave. New Orleans, LA 70118, U.S.A.
² Department of Logic Faculty of Arts Charles University Palachovo nám. 2 116 38 Praha 1, Czech Republic

Fundamenta Mathematicae, Tome 235 (2016) no. 2, pp. 153-166

Résumé

The ultrafilters on the partial order $([\omega ]^{\omega },\subseteq ^*)$ are the free ultrafilters on $\omega $, which constitute the space $\omega ^*$, the Stone–Čech remainder of $\omega $. If $U$ is an upperset of this partial order (i.e., a semifilter), then ultrafilters on $U$ correspond to closed subsets of $\omega ^*$ via Stone duality. If $U$ is large enough, then it is possible to get combinatorially nice ultrafilters on $U$ by generalizing the corresponding constructions for $[\omega ]^\omega $. In particular, if $U$ is co-meager then there are ultrafilters on $U$ that are weak $P$-filters (extending a result of Kunen). If $U$ is $G_\delta $ (and hence also co-meager) and $\mathfrak {d=c}$, then there are ultrafilters on $U$ that are $P$-filters (extending a result of Ketonen). For certain choices of $U$, these constructions have applications in dynamics, algebra, and combinatorics. Most notably, we give a new proof of the fact that there are minimal-maximal idempotents in $(\omega ^*,+)$. This was an outstanding open problem solved only recently by Zelenyuk.

Détail

DOI : 10.4064/fm182-2-2016

Keywords: ultrafilters partial order omega omega subseteq * ultrafilters nbsp omega which constitute space omega * stone ech remainder omega upperset partial order semifilter ultrafilters correspond closed subsets nbsp omega * via stone duality large enough possible get combinatorially nice ultrafilters generalizing corresponding constructions omega omega particular co meager there ultrafilters weak p filters extending result kunen delta hence co meager mathfrak there ultrafilters p filters extending result ketonen certain choices these constructions have applications dynamics algebra combinatorics notably proof there minimal maximal idempotents omega * outstanding problem solved only recently zelenyuk